Prove that similar matrices have the same rank

[zeion]

Homework Statement

Prove that similar matrices have the same rank.

Homework Equations

The Attempt at a Solution

Similar matrices are related via: B = P^-1AP, where B, A and P are nxn matrices..
since P is invertible, it rank(P) = n, and so since the main diagonal of P all > 0, multiplying by P will not change the rank of A, so rank B = rank A.

Is that seem right?

 

[VeeEight]

If you can show that rank(P^-1AP) is less than or equal to rank(A), then you are done since matrix similarity is symmetric (if A is similar to B, then B is similar to A).